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Ray based tomography using residual Stolt migration

Robert G. Clapp


In complex areas, residual vertical movement is not an effective method to calculate traveltime errors for image domain tomography. By scanning over velocity ratios using residual Stolt migration, a different criteria for a coherent image can be defined, and a traveltime error approximated. The resulting traveltime errors are more accurate, therefore the tomography procedure is more robust than the more traditional methodology. Results are shown on a complex 2-D dataset.

Depth migration is often necessary for complex structures, but requires an accurate interval velocity model. Estimating this velocity model is one of the essential problems in reflection seismology. One of the most common methods to estimate an interval velocity model is from ray-based reflection tomography after migration Stork (1992), which I will refer to as image domain tomography. In previous work Clapp and Biondi (2000); Clapp (2001), I showed how to use angle gathers Prucha et al. (1999); Sava and Fomel (2000) in conjunction with downward continuation based migration Biondi and Palacharla (1995); Gazdag and Sguazzero (1984) for back-projection. Post-migration reflection tomography is based on the fact that an offset gather (Kirchhoff) or angle gather (wavefield continuation) should be flat after migration. Deviation from flatness indicates a velocity error and can be converted into a travel time error and back-projected. Etgen (1990), among many others, pointed out that, in complex environments, looking at vertical moveout in gathers is not the optimal method to describe moveout errors. Biondi and Symes (2002) presented one alternative approach, constructing gathers where moveout is normal to an event.

Another approach is to use residual migration Fomel (1997); Levin et al. (1983); Rocca and Salvador (1982) to find the best focusing velocity. Stolt (1996) and Sava (1999a,b) showed how to do residual migration for wave continuation methods. These methods allow scanning over slowness field ratios. Audebert et al. (1996) noticed that when scaling the slowness field by a constant, ray behavior is unchanged. Audebert et al. (1997) described a method of updating the velocity model by back projecting along the normal ray.

In this paper I take these works a step further. After performing downward continuation based migration, I use residual migration to find a smooth field of $\gamma$ values that best focuses the data. I then convert these $\gamma$ values to approximate travel-time errors and back-project.

I begin by outlining a method of selecting my $\gamma$ field. I then describe the approximations used to convert $\gamma$ to $\bf \Delta t$. Finally, I show the procedure applied to a 2-D North Sea example.

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